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Applying Newton's 2nd to a rocket


Well-known member
Feb 1, 2012
A rockt with mass \(M\) is sitting motionless in space. The rocket is powered by steam with velocity \(V_s\) and mass per time \(C = \frac{dm}{dt}\). Write an equation for the rockets acceleration in terms of \(M, {} V_s\), and \(C\).

So \(\mathbf{F}_s = \mathbf{F}_r\) by Newton's 3rd. By Newton's 2nd,
\mathbf{F}_s &= \mathbf{F}_r\\
\frac{d(mV_s)}{dt} &= \frac{d(mV_r)}{dt}\\
m\frac{dV_s}{dt} + V_s\frac{dm}{dt} &= m\frac{dV_r}{dt} + V_r\frac{dm}{dt}\\
V_s\cdot C &= ma_r
Here we assumed the steam isn't accelerating and \(\frac{dm}{dt} = 0\) on the RHS. I understand assuming the steam acceleration is zero, but why is the \(\frac{dm}{dt}\) on the RHS zero and not \(C\)?